22 August 2023 marked a dark day in the world of statistics and data science, as we bid farewell to Professor C.R. Rao. As we mourn his demise, we reflect on the profound significance of his life's work in the context of our contemporary data-driven society. Professor Rao's contributions to statistics continue to shape the way we understand and interpret data in the modern world, and it is only fitting that we pay tribute to his enduring legacy.
A Life of Excellence
Calyampudi Radhakrishna Rao also known as Professor C.R. Rao began his journey in the world of statistics early in his life, and his passion for the subject never waned. Born on September 10, 1920, in India, his illustrious career spanned over eight decades, leaving an indelible mark on the field of statistics.
Behind every scientific advancement, there are pioneers who chart the course, breaking new ground and illuminating the path for generations to come. Professor C.R. Rao was undeniably one of these luminaries in the world of statistics. His work has not only propelled the field to remarkable heights but has also left an indelible mark on our daily lives, making the complex seem comprehensible and the challenging, manageable. Today, we pause to express our profound gratitude for his exceptional contributions. Let us delve into some of Professor Rao's groundbreaking work, which serves as the bedrock of numerous statistical concepts that shape our world.
A few of the statistical cornerstones resulted from his work include:
Cramér-Rao Bound:
Professor Rao, introduced a limit on the precision of unbiased parameter estimation as did Harald Cramer, independently, which is why this limit later came to be known as the Cramér-Rao bound. In his 1945 paper, Rao proved that any unbiased estimator of a parameter has a variance that is bounded below by the reciprocal of the Fisher information where Fisher information is the amount of information that a random sample X1, . . . , Xn contains about an underlying population parameter θ.
The Craḿer- Rao lower bound (CRLB) is ubiquitous in the signal processing literature and has found applications in almost every field of science and engineering. There are numerous extensions of the result including the quantum CRLB and the Bayesian CRLB. Dembo, Cover and Thomas provide a review of various inequalities in information theory and their connections to inequalities found in other areas of mathematics and physics. The Weyl-Heisenberg uncertainty principle-responsible for some of the most radical ideas in quantum mechanics for instance, can be derived from the Cram ́er- Rao inequality!
In modern applications such as machine learning and signal processing, the Cramér-Rao bound guides the design of estimation algorithms. It helps us understand the inherent limitations of parameter estimation, ensuring that our models are as accurate as possible given the available data.
Rao Blackwell Theorem:
The Rao blackwell theorem is a central result in statistical theory that ideated to improve the efficiency of estimators and consequently ours as well, if we’re being honest!
Formally, the result states that if g(X) is an estimator of an unknown parameter θ, then the estimator constructed by taking the conditional expectation of g(X) given the sufficient statistic T(X) has a smaller mean squared error (MSE). This essentially means that if we want an estimator with small MSE we can confine our search to estimators which are functions of the sufficient statistic.
Information Geometry:
The subject of information geometry blends several areas of statistics, computer science, physics, and mathematics. The subject evolved from the groundbreaking article published by legendary statistician C.R. Rao in 1945. His paper introduced the notion of distance or divergence between probability distributions parameterized by a q-dimensional vector θ. The paper was one of the earliest to apply differential geometric approaches to probability models, creating the framework for the new field of Information Geometry. In recent years, there has been tremendous interest in Information Geometry and its applications to Optimization, Machine Learning, and Deep Learning.
Other areas he worked in include multivariate analysis, estimation theory, and differential geometry, Linear Models, Design of Experiments and more! Rao Score test, Generalized inverse or as he called it, “pseudoinverse” of a singular matrix and many more such cornerstones of statistics that are just glimpses of the rich tapestry of Professor Rao's work.
Needless to say, he was also a visionary who understood that for Statistics to remain an influential discipline, it must adapt to the rapidly changing world, to remain a “powerful tool in acquiring knowledge in any field of enquiry.” Quoting Savage, Rao wrote: “The foundations of statistics are shifting, not only in the sense that they have always been, and will doubtless long continue to be, changing, but also in the idiomatic sense that no known system is quite solid.”
A scientist of extraordinary prescience, Rao predicted the Big Data revolution and the growth of new disciplines at the interface of mathematics, statistics, and computer science.
Prof. Rao’s work has been widely recognized with prestigious awards, reflecting the global impact of his contributions. In 2023, he was also awarded the International Prize in Statistics, an award often touted as the "statistics' equivalent of the Nobel Prize" for his monumental work 75 years ago that revolutionized statistical thinking. In announcing the award on April 1, the International Prize in Statistics noted that Rao’s “work more than 75 years ago continues to exert a profound influence on science.” It added: “In his remarkable 1945 paper published in the Bulletin of the Calcutta Mathematical Society, Calyampudi Radhakrishna (C.R.) Rao demonstrated three fundamental results that paved the way for the modern field of statistics and provided statistical tools heavily used in science today.”
The first result, known as the Cramer-Rao lower bound, provides a means for knowing when a method for estimating a quantity is as good as any method can be. The second result, named the Rao-Blackwell Theorem, provides a means for transforming an estimate into an optimal one. Taken together, the two methods form the foundation on which much of statistics is built.
The insights from Rao’s third result pioneered a new interdisciplinary field called “information geometry,” which has recently been used in Higgs boson measurements at the Large Hadron Collider, as well as in recent research on radars and antennas, along with contributing significantly to advancements in artificial intelligence, the announcement noted.
“In awarding this prize, we celebrate the monumental work by C.R. Rao, that not only revolutionized statistical thinking in its time, but also continues to exert enormous influence on human understanding of science across a wide spectrum of disciplines,” said Guy Nason, chair of the International Prize in Statistics Foundation. Not new to awards and accolades, the master statistician was also the recipient of Padma Vibhushan, US National Medal of Science, SS Bhatnagar Prize, and Guy Medal to name a few.
As we honour the life and mourn the demise of Professor C.R. Rao, we are reminded of the lasting legacy he leaves behind. C R Rao was the first batch student to receive the M.A. degree in Statistics from University of Calcutta. Immediately after getting the M.A. Degree, Rao joined ISI founded by the doyen of Indian statistics, P.C. Mahalanobis. C.R Rao played an important role, under the direction of Mahalanobis in setting up state statistical bureaus in different states of India and developing a network of statistical agencies at the district level for collection of data. Together with the Central Statistical Organization and the National Sample Survey in planning of which, C.R. Rao played a significant role, India has one of the best national statistical systems. He founded the Indian Econometric Society, which has been active in promoting quantitative studies in economics for planning purposes. After the demise of P C Mahalanobis, C R Rao worked as Secretary & Director of ISI from July 1972 to June 1976. He was Head and later Director of the Research and Training School at the Indian Statistical Institute for a period of over 40 years. It was during this time that Rao developed research and training programs and produced outstanding students and earned for ISI the name of Indian School of Statistics. During this period he also directed the training programs at the International Statistical Educational Center which led to the development of statistics in the South East Asian region. Rao was the Chairman of a UN Committee, which examined the demand for statistical personnel in Asian countries and recommended the establishment of an Institute for statistical development in South East Asia. On the basis of his recommendation The Asian Statistical Institute now known as Statistical Institute for Asia and Pacific was established in Tokyo to provide training to statisticians working in government and industrial organizations.
Hence, his contributions to the field of statistics are not only limited to his mathematical work but also in ways he has helped open up new demographics worldwide especially for India and the Indian budding minds. For which, we will be eternally grateful.
In a world where data is often described as the new oil, the legacy of Professor Rao serves as a guiding light for statisticians, data scientists, and researchers worldwide. His work reminds us of the power of statistics to unlock insights, make informed decisions, and ultimately improve our understanding of the complex systems that shape our lives.
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